MathDB

1

Part of 2010 Indonesia TST

Problems(10)

Nice sequence...

Source: 2nd Test - Indonesia TST IMO 2010 Training Camp P1

2/22/2017
Sequence un{u_n} is defined with u0=0,u1=13u_0=0,u_1=\frac{1}{3} and 23un=12(un+1+un−1)\frac{2}{3}u_n=\frac{1}{2}(u_{n+1}+u_{n-1}) ∀n=1,2,...\forall n=1,2,... Show that ∣un∣≤1|u_n|\leq1 ∀n∈N.\forall n\in\mathbb{N}.
Sequencealgebra
xf(y) - yf(x) = f (y/x), probably known

Source: 2010 Indonesia TST stage 2 test 1 p1

12/16/2020
Find all functions f:R→R f : R \to R that satisfies xf(y)−yf(x)=f(yx)xf(y) - yf(x)= f\left(\frac{y}{x}\right) for all x,y∈Rx, y \in R.
algebrafunctionalfunctional equation
Hard inequality

Source:

10/23/2010
Given a,b,c a,b, c positive real numbers satisfying a+b+c=1 a+b+c=1 . Prove that 1ab+bc+ca≥2a3(b+c)+2b3(c+a)+2c3(a+b)≥a+b+c \dfrac{1}{\sqrt{ab+bc+ca}}\ge \sqrt{\dfrac{2a}{3(b+c)}} +\sqrt{\dfrac{2b}{3(c+a)}} + \sqrt{\dfrac{2c}{3(a+b)}} \ge \sqrt{a} +\sqrt{b}+\sqrt{c}
inequalities
Indonesian IMO TST 2010 Test 4 problem 1

Source:

1/8/2011
find all pairs of relatively prime natural numbers (m,n) (m,n) in such a way that there exists non constant polynomial f satisfying gcd(a+b+1,mf(a)+nf(b)>1 gcd(a+b+1, mf(a)+nf(b) > 1 for every natural numbers a a and b b
algebrapolynomialnumber theoryrelatively primenumber theory proposed
3x3 system x^5 = 2y^3 + y - 2, y^5 = 2z^3 + z - 2, z^5 = 2x^3 + x - 2

Source: 2010 Indonesia TST stage 2 test 5 p1

12/16/2020
Find all triplets of real numbers (x,y,z)(x, y, z) that satisfies the system of equations x5=2y3+y−2x^5 = 2y^3 + y - 2 y5=2z3+z−2y^5 = 2z^3 + z - 2 z5=2x3+x−2z^5 = 2x^3 + x - 2
algebrasystem of equations
a,b,c,x,y,z in inequality

Source: Indonesia IMO 2010 TST, Stage 1, Test 1, Problem 1

11/12/2009
Let a a, b b, and c c be non-negative real numbers and let x x, y y, and z z be positive real numbers such that a\plus{}b\plus{}c\equal{}x\plus{}y\plus{}z. Prove that \dfrac{a^3}{x^2}\plus{}\dfrac{b^3}{y^2}\plus{}\dfrac{c^3}{z^2} \ge a\plus{}b\plus{}c. Hery Susanto, Malang
inequalitiesinequalities proposed
geometry biimplication

Source: Indonesia IMO 2010 TST, Stage 1, Test 2, Problem 1

11/12/2009
Let ABCD ABCD be a trapezoid such that AB∥CD AB \parallel CD and assume that there are points E E on the line outside the segment BC BC and F F on the segment AD AD such that \angle DAE \equal{} \angle CBF. Let I,J,K I,J,K respectively be the intersection of line EF EF and line CD CD, the intersection of line EF EF and line AB AB, and the midpoint of segment EF EF. Prove that K K is on the circumcircle of triangle CDJ CDJ if and only if I I is on the circumcircle of triangle ABK ABK. Utari Wijayanti, Bandung
geometrytrapezoidcircumcirclegeometry proposed
2009 divides f(c) for some c

Source: Indonesia IMO 2010 TST, Stage 1, Test 3, Problem 1

11/12/2009
Let f f be a polynomial with integer coefficients. Assume that there exists integers a a and b b such that f(a)\equal{}41 and f(b)\equal{}49. Prove that there exists an integer c c such that 2009 2009 divides f(c) f(c). Nanang Susyanto, Jogjakarta
algebrapolynomialnumber theory proposednumber theory
1,2,...,20 on the blackboard

Source: Indonesia IMO 2010 TST, Stage 1, Test 4, Problem 1

11/12/2009
The integers 1,2,…,20 1,2,\dots,20 are written on the blackboard. Consider the following operation as one step: choose two integers a a and b b such that a\minus{}b \ge 2 and replace them with a\minus{}1 and b\plus{}1. Please, determine the maximum number of steps that can be done. Yudi Satria, Jakarta
invariantcombinatorics proposedcombinatorics
geometry and number theory

Source: Indonesia IMO 2010 TST, Stage 1, Test 5, Problem 1

11/12/2009
Is there a triangle with angles in ratio of 1:2:4 1: 2: 4 and the length of its sides are integers with at least one of them is a prime number? Nanang Susyanto, Jogjakarta
geometryratiotrigonometryalgebrapolynomialinductionfunction